The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse…

Equation of auxiliary circle is

$x^2+y^2=9 ..\left(\mathrm{i}\right)$

$\therefore$ Equation of $AM$ is $\frac{x}{3}+\frac{y}{1}=1 ...\left(\mathrm{ii}\right)$

On solving Eqs. (i) and (ii), we get

$M\left(-\frac{12}{5},\frac{9}{5}\right)$

Now, area of $\mathrm{\Delta }AOM=\frac{1}{2}OA\times MN=\frac{27}{10}$ sq units

Hence, option (d) is correct.